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Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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02 Jun 2014, 08:35

Hello from the GMAT Club BumpBot!

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Each week we'll be posting several questions from The Official Guide for GMAT® Review, 13th Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

We'll be glad if you participate in development of this project: 1. Please provide your solutions to the questions; 2. Please vote for the best solutions by pressing Kudos button; 3. Please vote for the questions themselves by pressing Kudos button; 4. Please share your views on difficulty level of the questions, so that we have most precise evaluation.

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I can suggest two solutions:

Solution A

We can compute the sum of the angles from the five triangles created on the sides of the pentagon ABCDE. In those triangles, we have five pairs of congruent angles (see them marked by colored arcs in the attached drawing). Those angles are external angles for the pentagon and their sum is \(360^o\). See at the end of the post the justification for the fact that in every convex polygon, the sum of the external angles is \(360^o\). Therefore, v + x + y + z + w = 5 ∙ 180 – 2 ∙ 360 = 900 – 720= 180.

Solution B

Since the question is a multiple choice one, we can assume that there is one correct answer and that that answer does not depend on the shape of the “star”. Assuming that the star can be inscribed in a circle, we can see that the requested sum of the angles is 360/2 = 180, because each angle is inscribed in the circle and the five corresponding arcs complete the circle. Remark: If one of the answers would have been “It cannot be determined” or something similar, than this argument wouldn’t work.

Correct answer: C

Sum of the external angles for a convex polygon:

We know that the sum of the interior angles in a convex polygon with n sides (n being a positive integer greater than 2) is given by the formula: (n – 2)∙180 = 180n – 360. Each external angle is 180 – the corresponding interior angle. Therefore, the total sum of the exterior angles is 180n – (n – 2) ∙ 180 = 180n – 180n + 360 = 360.

Note: Convex polygons have the property that each of their angles is less than 180. All the polygons dealt with on GMAT are convex (triangle, quadrilateral, pentagon, hexagon,...) or are made up of convex polygons. In this question, the figure of the star, without the sides of the small convex pentagon, is an example of a non-convex decagon: it has 10 sides, and 5 angles which are greater, and 5, which are smaller than 180.

I don't get the equation of solution A... can anyone shed some light?

Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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01 Jul 2014, 04:17

Attachment:

star-sol.png [ 10.08 KiB | Viewed 1272 times ]

i didn't know the central angle theorem, and tried solving with a different way. sorry for my mad paint skills :D

here is my solution: 1. Draw a line via vertex of angle Y, parallel to the line between angles V and Z. In a picture the red coloured lines are parallel. 2. Draw a line via vertex of angle Y, parallel to the line between angles X and Z (violet coloured) 3. Draw a line via vertex of angle Y, parallel to the line between angles X and W (blue coloured)

The following can be concluded from the pic accoridng to thales theorem: a. angles between red and violet lines will be same (angle Z) b. angles between black and red lines will be same (angle V) c. angles between blue and black lines will be same (angle W) d. angles between blue and violet lines will be same (angle X)

as a result sum of 5 angles will be a violet line and equal to 180 degrees

My solution is obviously not as simple and quick as Bunuel's one, but maybe you can use my approach for solving similar problems

Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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19 Nov 2014, 08:54

Bunuel wrote:

Let's simplify the problem by imagining that we have a star that is inscribed in a circle as shown below:

Hi Bunuel. How can we assume that the star could be inscribed within a circle in the absence of information. Further, the general GMAT assumption is that all diagrams are not drawn to scale unless the contrary is mentioned.

Am I missing anything here? Please help!
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Cheers!!

JA If you like my post, let me know. Give me a kudos!

Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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19 Nov 2014, 10:21

3

This post received KUDOS

Wow! See what I found with a little googling!

A star is always regularly shaped (this clarifies why Bunuel assumed that the start would get inscribed in the circle)! 1. The sum of the angles formed at the tips of the five pointed star is 180; the sum of the angles formed at the tips of the six pointed star is 360. 2. The formula for the sum of the angle measurements at the tips of an n-pointed star is f(n)=180(n)-720 where n is an integer greater than 4.

Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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15 Dec 2014, 22:57

Hi Bunuel, can you please elaborate how you made the assumption that "Let's simplify the problem by imagining that we have a star that is inscribed in a circle as shown below"?

Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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15 Dec 2014, 22:57

Hi Bunuel, can you please elaborate how you made the assumption that "Let's simplify the problem by imagining that we have a star that is inscribed in a circle as shown below"?

Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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27 Aug 2015, 17:08

1

This post received KUDOS

Interior angle of a polygon = \(((n-2)180)/n\) For Pentagon = \((5-3)180/5\) = 108 sum of angles in each triangle angle x+(180-108)+(180-108) = 180 .... x = 36 there are 5 triangles ... 36*5 = 180 Answer (C)

Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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16 Mar 2016, 06:48

Alternate Solution with just the basics. (I have not named every vertex separately in order to avoid confusion (mess). Every vertex of the star is named by the angle it depicts in the picture)

Attachments

File comment: y+A+B+C = 360 (sum of all the angles in quadrilateral-YABC ) A= 180-(y+v) (sum of all angles in a triangle is 180,Triangle YAV) B= 180-(x+z) (Triangle XBZ) C= 180-(y+w) (Triangle YCW)

Now, Substitute the values of A, B and C in the equation : A+B+C+y=360 (180-y-v) + (180-x-z) + (180-y-w) + y =360 By solving the above, we get: 540 - w - v - z - x - 2y + y=360 x+y+v+z+w=180 ANSWER - C

Re: In the figure shown, what is the value of v+x+y+z+w? [#permalink]

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05 Nov 2016, 14:29

I think the easiest way to approach this problem was assuming we were dealing with a regular pentagon (all angles equal) and figuring out what each angle was using the interior angle formula --> (5-1)(180) = 540 --> 540/# of angles = 108

We know that the angle outside, opposite the interior angle of the pentagon is the same, therefore the two adjacent triangle angles will be 360-216 = 144. Divide by 2 to get them symmetrical and you will find each triangle off the pentagon has two angles that are 72 degrees, making the third of each 36.

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